Collective cell motility is an important aspect of several developmental and pathophysiological processes. Despite its importance, the mechanisms that allow cells to be both motile and adhere to one another are poorly understood. In this study we establish statistical properties of the random streaming behavior of endothelial monolayer cultures. To understand the reported empirical findings, we expand the widely used cellular Potts model to include active cell motility. For spontaneous directed motility we assume a positive feedback between cell displacements and cell polarity. The resulting model is studied with computer simulations and is shown to exhibit behavior compatible with experimental findings. In particular, in monolayer cultures both the speed and persistence of cell motion decreases, transient cell chains move together as groups and velocity correlations extend over several cell diameters. As active cell motility is ubiquitous both in vitro and in vivo, our model is expected to be a generally applicable representation of cellular behavior.